What was the most interesting part of the material for you?
I am excited to do proofs about calculus. I loved the calculus classes I took, and it will be fun to more rigorously understand the concepts.
What was the most difficult part of the material for you?
I have a hard time figuring out what to use for N in the proofs. As we do more practice problems it will make more sense.
Saturday, November 29, 2014
Monday, November 24, 2014
Responses to Questions, due on November 25
What have you learned in this course?
In this course, I've learned to become more rigorous in my mathematical understanding. In calculus we focused less on rigor, so this class has been useful in that regard. I have become much more precise and articulate when communicating mathematical ideas.
How might these things be useful to you in the future?
The mathematical rigor I've gained from this class will be essential in the upper-level math courses I take in the ACME program. Being able to show something is true, instead of just applying given theorems, gives a much deeper understanding of the idea.
In this course, I've learned to become more rigorous in my mathematical understanding. In calculus we focused less on rigor, so this class has been useful in that regard. I have become much more precise and articulate when communicating mathematical ideas.
How might these things be useful to you in the future?
The mathematical rigor I've gained from this class will be essential in the upper-level math courses I take in the ACME program. Being able to show something is true, instead of just applying given theorems, gives a much deeper understanding of the idea.
Sunday, November 23, 2014
11.5-11.6, due on November 23
What was the most interesting part of the material for you?
I think the fundamental theorem of arithmetic is really cool, because it makes so much sense and makes comparing numbers much easier. Prime factorization is interesting.
What was the most difficult part of the material for you?
I had a hard time understanding the proof for Corollary 15. Using induction for this doesn't make sense to me, because I don't see why we know the base case works.
I think the fundamental theorem of arithmetic is really cool, because it makes so much sense and makes comparing numbers much easier. Prime factorization is interesting.
What was the most difficult part of the material for you?
I had a hard time understanding the proof for Corollary 15. Using induction for this doesn't make sense to me, because I don't see why we know the base case works.
Thursday, November 20, 2014
Pre-Test post, due on November 21
Which topics and theorems do you think are the most important out of those we have studied?
The most important topics and theorems probably include: denumerability, countability, cardinality, the Schröder-Bernstein Theorem, the equivalence of the GCD and the least linear combination of two numbers, and the division and Euclidean algorithms.
What kinds of questions do you expect to see on the exam?
I expect to be asked to prove several theorems about various topics covered in this section. I expect to be asked to show that two sets are of equal or unequal cardinality, both for finite and infinite sets. I need to be able to show that numbers of varying definitions are either composite or prime.
What do you need to work on understanding better before the exam? Come up with a mathematical question you would like to see answered or a problem you would like to see worked out.
I need to practice the Euclidean algorithm and go over the proof of that method, so I can better understand why it works. I need to go over proving statements comparing the cardinality of sets. I need to go over the proofs for the theorems we might need to prove on the test.
The most important topics and theorems probably include: denumerability, countability, cardinality, the Schröder-Bernstein Theorem, the equivalence of the GCD and the least linear combination of two numbers, and the division and Euclidean algorithms.
What kinds of questions do you expect to see on the exam?
I expect to be asked to prove several theorems about various topics covered in this section. I expect to be asked to show that two sets are of equal or unequal cardinality, both for finite and infinite sets. I need to be able to show that numbers of varying definitions are either composite or prime.
What do you need to work on understanding better before the exam? Come up with a mathematical question you would like to see answered or a problem you would like to see worked out.
I need to practice the Euclidean algorithm and go over the proof of that method, so I can better understand why it works. I need to go over proving statements comparing the cardinality of sets. I need to go over the proofs for the theorems we might need to prove on the test.
Tuesday, November 18, 2014
11.3-11.4, due on November 19
What was the most interesting part of the material for you?
It seems like Theorem 11.7 makes it much easier to find greatest common denominators. It seems like it would be easy to plot and find three-dimensional minima for the equation for linear combination. Why didn't we learn this in high school algebra?
What was the most difficult part of the material for you?
I have a hard time understanding Lemma 11.9. I don't see why the Euclidean Algorithm works the way it does.
It seems like Theorem 11.7 makes it much easier to find greatest common denominators. It seems like it would be easy to plot and find three-dimensional minima for the equation for linear combination. Why didn't we learn this in high school algebra?
What was the most difficult part of the material for you?
I have a hard time understanding Lemma 11.9. I don't see why the Euclidean Algorithm works the way it does.
Saturday, November 15, 2014
11.1-11.2, due on November 17
What is the most interesting part of the material for you?
I think it's cool that we get to study number theory. I've always heard from math teachers in high school about number theory but I never understood what it was or had the chance to study it.
What was the most difficult part of the material for you?
I had a hard time following the proof of the Division Algorithm. Specifically, it didn't quite make sense how the book shows that q and r are a unique set of a quotient and a remainder for an integer.
I think it's cool that we get to study number theory. I've always heard from math teachers in high school about number theory but I never understood what it was or had the chance to study it.
What was the most difficult part of the material for you?
I had a hard time following the proof of the Division Algorithm. Specifically, it didn't quite make sense how the book shows that q and r are a unique set of a quotient and a remainder for an integer.
Thursday, November 13, 2014
The rest of 10.5, due on November 14
What was the most interesting part of the material for you?
It was fun to read about how the various theorems in this section were proved in pieces and then completely over a long period of time. Also, it makes a lot of sense that the power set of the natural numbers and the set of real numbers are numerically equivalent, so it's cool to see that that's true.
What was the most difficult part of the material for you?
I have a hard time understanding the Axiom of Choice. It seems rather trivial to my understanding, so I'm sure I don't quite understand what the statement means. I'll enjoy talking about it in class today.
It was fun to read about how the various theorems in this section were proved in pieces and then completely over a long period of time. Also, it makes a lot of sense that the power set of the natural numbers and the set of real numbers are numerically equivalent, so it's cool to see that that's true.
What was the most difficult part of the material for you?
I have a hard time understanding the Axiom of Choice. It seems rather trivial to my understanding, so I'm sure I don't quite understand what the statement means. I'll enjoy talking about it in class today.
Tuesday, November 11, 2014
10.5 up to Theorem 10.18, due on November 12
What was the most interesting part of the material for you?
I think the Schröder-Bernstein Theorem is cool, because it allows us more flexibility in making comparisons between the sizes of infinite sets. This is very similar to showing sets are equal by showing they are subsets of each other.
What was the most difficult part of the material for you?
I don't quite yet understand the proof for the Schröder-Bernstein Theorem. When we go over it in class, I'm sure it will make more sense. The proofs for the other theorems in this section make sense to me.
I think the Schröder-Bernstein Theorem is cool, because it allows us more flexibility in making comparisons between the sizes of infinite sets. This is very similar to showing sets are equal by showing they are subsets of each other.
What was the most difficult part of the material for you?
I don't quite yet understand the proof for the Schröder-Bernstein Theorem. When we go over it in class, I'm sure it will make more sense. The proofs for the other theorems in this section make sense to me.
Saturday, November 8, 2014
10.4, due on November 10
What is the most interesting part of the material for you?
The Continuum Hypothesis is sort of mind-blowing. Even more mind-blowing is the fact that it's been shown to be impossible to prove or disprove. I don't understand how that works, but it would be cool to find out.
What is the most difficult part of the material for you?
I didn't understand the second half of the proof of Theorem 10.15. I got a little confused by all the seemingly arbitrary set definitions used in the proof. We'll talk about it in class and it will make more sense when we walk through it.
The Continuum Hypothesis is sort of mind-blowing. Even more mind-blowing is the fact that it's been shown to be impossible to prove or disprove. I don't understand how that works, but it would be cool to find out.
What is the most difficult part of the material for you?
I didn't understand the second half of the proof of Theorem 10.15. I got a little confused by all the seemingly arbitrary set definitions used in the proof. We'll talk about it in class and it will make more sense when we walk through it.
Friday, November 7, 2014
10.3, due on November 7
What was the most difficult part of the material for you?
I was a little confused by the first proof in the section that talked about how (0,1) in the real numbers is uncountable. It seems like all we did was give an irrational number as proof.
What was the most interesting part of the material for you?
I think it's cool that (0,1) in the real numbers and the set of all real numbers are numerically equivalent. This is analogous to how the natural numbers are numerically equivalent to the integers. In both examples, it seems like the first set is of a smaller size than the second, but it's not.
I was a little confused by the first proof in the section that talked about how (0,1) in the real numbers is uncountable. It seems like all we did was give an irrational number as proof.
What was the most interesting part of the material for you?
I think it's cool that (0,1) in the real numbers and the set of all real numbers are numerically equivalent. This is analogous to how the natural numbers are numerically equivalent to the integers. In both examples, it seems like the first set is of a smaller size than the second, but it's not.
Tuesday, November 4, 2014
10.2, due on November 5
What was the most difficult part of the material for you?
I had a hard time understanding why the crossing of two denumerable sets is also denumerable. It seems like that's kind of like taking infinity times infinity. But after looking at the proof it makes more sense.
What was the most interesting part of the material for you?
I think it's way cool that we can look at numerical equivalence for sets that are infinite. It kind of reminds me of using l'Hospital's Rule to simplify limits with infinity on top and bottom. It seems like you're sort of comparing the "size" of the top and bottom of the limit.
I had a hard time understanding why the crossing of two denumerable sets is also denumerable. It seems like that's kind of like taking infinity times infinity. But after looking at the proof it makes more sense.
What was the most interesting part of the material for you?
I think it's way cool that we can look at numerical equivalence for sets that are infinite. It kind of reminds me of using l'Hospital's Rule to simplify limits with infinity on top and bottom. It seems like you're sort of comparing the "size" of the top and bottom of the limit.
Saturday, November 1, 2014
10.1, due on November 3
What was the most difficult part of the material for you?
I don't really understand why Theorem 10.1 is important. I can see how it's important to be able to show that you can create a bijective function between to numerically equivalent sets, but I don't see why we need to know that this relation is an equivalence relation.
What was the most interesting part of the material for you?
I completely understand the part from above now! It seems like the point of the theorem is to show that numerical equivalence is reflexive, symmetric, and transitive! It totally makes sense after reading it again.
I don't really understand why Theorem 10.1 is important. I can see how it's important to be able to show that you can create a bijective function between to numerically equivalent sets, but I don't see why we need to know that this relation is an equivalence relation.
What was the most interesting part of the material for you?
I completely understand the part from above now! It seems like the point of the theorem is to show that numerical equivalence is reflexive, symmetric, and transitive! It totally makes sense after reading it again.
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